0
$\begingroup$

Let $\emptyset \neq A \subseteq \mathcal{S}$ on a state space $\mathcal{S}$ of a Markov chain $\{X_n\}_{n=1}^\infty$. We define

$$T_A := \inf\{n \geq 1 \mid X_n \in A\}$$

with $\inf \emptyset = +\infty$?

How do I interpret this infinum? In particular, what does the $X_n \in A$ mean?

Is it short written for

$$T_A = \inf\{n \geq 1 \mid X_n(\Omega) \cap A \neq\emptyset\}$$ if we work on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$?

Is $T_A$ a random variable? I see that one calculates probabilities like $\mathbb{P}(T_y < \infty)$ and expectations $\mathbb{E}T_y$ so I think it should be a random variable.

Or maybe, it is

$$T_A(\omega) := \inf \{n \geq 1\mid X_n(\omega) \in A\}$$ for $\omega \in \Omega$? Can someone clarify?

$\endgroup$
0
$\begingroup$

$T_A$ is a random variable (more precisely a stopping time) defined by $$ \forall\omega\in\Omega,\quad T_A(\omega)=\inf\{n\ge1\mid X_n(\omega)\in A\}. $$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy